The “barbell” portfolio has long been considered an investment strategy. Since his fame after the 2010 publication of the book The Black Swan, Nassim Nicholas Taleb has often been associated with the strategy. Recent research illustrates the critical connection between the barbell, core-satellite portfolios and safety-first financial planning – and how advisors can improve on using standard deviation as a measure of risk.
That research is in an article, “Tail Risk Constraints and Maximum Entropy,” that appeared in the journal Entropy, coauthored by Donald Geman, Hélyette Geman, and Taleb. Its thesis, approach and results are worth a closer look. The basic idea is to anchor a portfolio to a hard, downside-only risk constraint – i.e., not standard deviation – and then to split the portfolio into two elements that meet two very different objectives: a hedged element that strictly meets the downside risk constraint and an aggressive element in which you maximize the uncertainty of the result.
Don Geman was a fellow graduate student when I was a PhD student in the pure mathematics department at Northwestern University. We had not been in touch since graduate school. I had vaguely heard of his success as a mathematician, and that he was based at Johns Hopkins University and France’s École Normale Supérieure. What I was not aware of was that Don’s wife, Hélyette, also a distinguished mathematician, was the supervisor of Taleb’s PhD thesis.
The basic idea of the barbell
The basic idea of the barbell strategy is that the portfolio is divided into two parts: an extremely conservative portfolio for safety, and a highly speculative portfolio for extra rewards with extra risk. A strategy that has sometimes been placed in this category – though a bit of a bastardization – is the core-and-satellite strategy, in which the bulk of an equity market portfolio is placed in a passive index fund, or funds, while the smaller portion, the satellite, speculates in concentrated active funds or hedge funds.
In the Gemans and Taleb’s article, motivation is provided by their claim that the measure of risk in nearly all investment finance theory, standard deviation of return, is unsatisfactory because its minimization not only limits downside risk, but also curtails upside opportunity. Better, they say, to impose a risk-control criterion that only limits downside risk in a manner that accords with the specific risks that investors want to avoid.
The risk-control criterion that the authors choose is a combination of VaR and CVaR. VaR – “value at risk” – is the portfolio loss that could occur during a specified time period with a probability p. “For example,” says a CFA Institute publication, “a portfolio that is expected to lose no more than $1 million 95% of the time (or 19 of every 20 days) has a VaR of $1 million.”
Note that VaR tells you how likely it is that you will lose no more than $1 million, but it doesn’t tell you how much more than $1 million you might lose. VaR has been criticized for not providing this information. This is important, because if there is a 5% chance that $1 million will be lost during the next 20 days, but the loss will be due to a rogue trader, that loss could actually be $5 billion.
CVaR, or “conditional value at risk,” is meant to compensate for VaR’s missing information. CVaR is the expected loss given (i.e. conditional on) the fact that at least $1 million is lost. So if, for example, a portfolio has a VaR of $1 million at the 95% confidence level, but the 5% tail event consists of a loss of $5 billion, then CVaR is $5 billion, because the expected loss given that a loss greater than $1 million occurs is $5 billion.
The authors’ approach to constraining risk
With VaR and CVaR as their risk measures, the authors then define the portfolio selection problem. This is very different, of course, from the Markowitz problem, which is to maximize portfolio expected return given the portfolio’s standard deviation (or variance) of return.
The solution to the problem can then take two forms, depending on what assumptions are made about the probability distributions of asset returns. The authors are loath to assume normal distributions of returns – unsurprisingly given Taleb’s inclination toward fat tails and the assumption that even probabilities are uncertain, especially of tail events – but they tackle that case anyway, because it is the easier case.
Their methodology is to constrain VaR and CVaR, and then ask what portfolio meets those constraints. Or if more than one portfolio meets those constraints, then which of those portfolios satisfies another constraint, such as a specified expected return, or maximizes another variable.
Normal probability distributions
In the case of normal probability distributions of portfolio returns, the answer is simple. If values of the VaR and CVaR constraints are given (and they are not degenerate, for example both VaR and CVaR are $1 million), then there is only one normal distribution that will satisfy them. That is to say, there is only one pair of parameters – expected return and standard deviation – of the normal distribution of portfolio returns that will satisfy the constraints on VaR and CVaR.
If you know the expected returns, standard deviations and correlation coefficients of the constituent assets, which you would need to know anyway to calculate the portfolio on the Markowitz efficient frontier, then you can use them to calculate the portfolio (or portfolios) that has the required expected return and standard deviation, and thus satisfies the VaR and CVaR constraints.
However, in order to do this you need to be careful about choosing the VaR and CVaR constraints. For some pairs of constraints, the result will be that the portfolio’s expected return is negative. This is not what is desired, so the VaR and CVaR need to be chosen so that the expected return is positive. In order to do this, VaR and CVaR can’t constrain the risk too much.
Unknown multivariate distributions
The authors plainly prefer to consider unknown probability distributions of returns, though there still has to be something known about them. For these cases, they say that “these methods can be made robust using constructions that, upon paying an insurance price, no longer depend on parametric assumptions. This can be done using derivative contracts or by organic construction…”
This statement is easiest to explain for stop-loss constraints, which is to say, when VaR and CVaR are the same. (As noted above, this can’t be satisfied if the returns distribution is normal.) For example, “paying an insurance price” means that under those circumstances a put option purchased on a suitable portfolio will meet the constraint.
By “organic construction” they mean that, for example, you could put 80% of your portfolio in numéraire securities (such as U.S. Treasury bonds), and then the risk of losing more than 20% of your portfolio would be zero (assuming you buy and hold; rebalancing doesn’t work).
If you don’t know what kind of probability distributions you’re dealing with, then just specifying VaR and CVaR doesn’t determine what they are, as it does when you know you’re dealing with normal distributions. In fact, it leaves an infinity of possible distributions that will all satisfy the VaR and CVaR constraints.
How, then, to choose among these probability distributions? This is where it gets interesting. The authors impose the additional condition that the risky part of the barbell portfolio must “maximize uncertainty.”
The principle is that once you’ve secured a level of safety in the conservative part of the portfolio (which you might have done using a put option, or Treasury bonds, or some other measure), then the rest is play money. You might as well play as fast and loose with it as you can – though subject again to some constraint, such as the expected return on that part of the portfolio.
Entropy is generally considered a measure of uncertainty. It is the expected value of the negative of the logarithm (to the base 2) of the probabilities in a probability distribution.
For example, for the toss of a fair coin (½ probability of heads, ½ of tails), the entropy is one, because the expected information you get from the result – that is, from resolving the uncertainty – is one bit (the “bit” is the unit of entropy). If the coin is not fair, however, but has a probability of 25% for tails and 75% for heads then the entropy (uncertainty) is less than one – specifically it is 0.811278 – because there’s less uncertainty about the result when the coin is loaded.
The Gemans and Taleb take as their program to maximize the entropy of the portfolio’s probability distribution of returns, given the constraints VaR and CVaR. Fortunately, given one or two more constraints on those returns – such as their expectation – the probability distribution that maximizes entropy can be found. Using some interesting mathematics that I won’t go into, they derive properties of that distribution. Because there is so much leeway for choosing the type of probability distribution, they can – and do – choose a case that has fat tails and another that does not.
Correspondence to a “safety net” strategy
Some advisors, consultants and commentators (myself and my coauthors of our 2014 book, The 3 Simple Rules of Investing, included) have suggested the possibility of a “safety net” strategy. The idea, for an investor that is an individual or family, is to determine a “bare minimum” future income that they could live with if absolutely necessary, and lock that in with a very conservative portfolio such as a single-premium immediate annuity (SPIA) and/or a ladder of Treasury bonds. Given that safety net, the investor is then assumed immunized against total panic, and therefore able to invest the rest of the portfolio in risky assets. This strategy is, therefore, similar to – or a variant of – a barbell strategy.
An important question is, does such a strategy pay too much for too absolute a level of protection (and therefore sacrifice too much upside)? Analogies abound. How much is it worth paying to make your travel by automobile absolutely safe? Should you drive a Sherman tank? Is it worth it? How much to make commercial flights absolutely safe from terrorists? How much to buy extremely deep defense-in-depth safety for nuclear reactors? Enough to make their cost uncompetitive with other forms of energy?
This is a very difficult question to answer conclusively. There is no clear method of cost-benefit analysis that can answer it. Along with the safety benefit of the safety-net approach, one important benefit is that it should drastically reduce the investor’s vulnerability to panic. A stock-bond portfolio with a similar expected return under rebalancing may not reduce that vulnerability as much. But this is something that cannot be entirely decided quantitatively – though some analyses can help. A good example is Joe Tomlinson’s analysis of combining SPIAs with the smoothing of year-to-year withdrawals.
Whatever is the right way to look at how to control risk in a portfolio (and there can never be only one “right” way, because people’s concepts of risk are kaleidoscopic), it is a breath of fresh air to read a paper that doesn’t automatically assume that risk is the standard deviation of a portfolio’s returns distribution. We need, in order to give meaningful practical advice on investing in real-world circumstances, to break free of the straitjacket of modern portfolio theory.
Michael Edesess is a mathematician and economist, a senior research fellow with the Centre for Systems Informatics Engineering at City University of Hong Kong, chief investment strategist of Compendium Finance and a research associate at EDHEC-Risk Institute. In 2007, he authored a book about the investment services industry titled The Big Investment Lie, published by Berrett-Koehler. His new book, The Three Simple Rules of Investing, co-authored with Kwok L. Tsui, Carol Fabbri and George Peacock, was published by Berrett-Koehler in June 2014.
